A one-dimensional C^3 example shows that vanishing-viscosity limits for mechanical Hamiltonian ergodic problems may not exist.
Agmon, Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-Body Schrodinger Operations, (MN-29)
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Nonexistence of vanishing-viscosity limits for mechanical Hamiltonian ergodic problems
A one-dimensional C^3 example shows that vanishing-viscosity limits for mechanical Hamiltonian ergodic problems may not exist.